Prepare Lucille's Smokehouse BBQ Garlic Mayonnaise according to the recipe in the Box below. Set aside.
Trim bottom outer leaves from artichokes. Cut off any spiny leaf tips with kitchen sheers. Peel stems.
Cut artichokes in 1/2. With a small, sharp knife, scoop out and discard the hairy chokes. Set aside.
In a large bowl, add olive oil, tarragon, lemon juice and salt. Whisk to mix well.
Add artichoke halves to marinade. Marinate at room temperature for 1 hour.
Remove from bowl. Reserve marinade.
Heat grill to high direct heat.
When grill is hot, place artichoke halves in center of cooking grate. Grill until tender, about 20 to 25 minutes, basting frequently with reserved olive oil marinade.
Remove from grill. Serve hot with any remaining marinade and garlic mayonnaise.